Statistics for Environmental Engineers

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z=

(pi — p2) — 0

2p(1 — p)

where p = (p1 + p2)/2.

Fleiss (1981) suggests that when n is small (e.g., n < 20), the computation of z should be modified to:

(pi — p2) — —

z=

2 p( 1 — p)

The (—1/n) in the numerator is the Yates continuity correction, which takes into account the fact that a continuous distribution (the normal) is being used to represent the discrete binomial distribution of sample proportions. For reasonably small n (e.g., n = 20), the correction is -1/n = -0.05, which can be substantial relative to the differences usually observed (e.g., p1 — p2 = 0.15). Not everyone uses this correction (Rosner, 1990). If it is omitted, there is a reduced probability of detecting a difference in rates. As n becomes large, this correction factor becomes negligible because the normal distribution very closely approximates the binomial distribution.

Case Study Solution

Eighty organisms (n1 = n1 = 80) were exposed to each treatment condition (control and effluent) and the sample survival proportions are observed to be:

p1 = 72/80 = 0.90 and    p2 = 64/80 = 0.80

We would like to use the normal approximation to the binomial distribution. Checking to see whether this is appropriate gives:

and    n( 1 — pj ) = 80( 0.10 ) = 8

and    n( 1 — p2) = 80( 0.20) = 16

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